Time is smooth, probably. I mean, last time I checked, the universe doesn’t lag. There haven’t been any drops in the universal framerate. Actually, there doesn’t appear to be a universal framerate. If there was, would that mean the present has a fixed duration? That “now” is one frame? That events happening now take time before solidifying into the past? If there was, we couldn’t possibly know, right? I mean, we’re just the universe observing itself, so we follow that same framerate. And if we ever lag, we won’t notice. The machine running the universe at 6 gajillion fps would just fill in the details, and we would never be the wiser.
So, time is smooth, probably. At least, that’s what the physicists tell me, so I’ll pretend like our current understanding of the universe is correct until someone comes along with a better idea. Not only is time smooth, but it feels constant to us. Obviously, my time may appear faster or slower than your time because… well, relativity and all that… but my time feels constant to me, and your time feels constant to you. It’s a reassuring thought, that everything will continue moving forward at the same pace. It was a reassuring thought, until H. G. Wells came along and said we “are passing along the Time-Dimension with a uniform velocity from the cradle to the grave.” Gee, thanks Herbert. Really killed the vibe.
Anyway, uniform velocity time is an interesting idea. If time is moving forward at a constant rate, and your temporal position is just the time, then your location is just the time.
$$p_1(t)=t$$
Here, I drew a picture for you.
“What happens at time equals 0?” you might ask. “It doesn’t really matter,” I might reply. “What are the units?” you might ask. “It doesn’t really matter?” I might reply. “Well, I’m in my timezone, but what happens when you have another timezone?” you might ask.
“You get two lines.” I might reply.
“Ha! You fool!” you might call me. “You’ve fallen right into my trap! Now I can time travel using timezones.”
“Okay, but what’s the time while you’re in the plane?” I’ll ask.
“I’ll just skip ahead when I land.” You’ll reply.
“But then time isn’t smooth.” I’ll say.
“Then I’ll draw a line between the two lines.”
“But then time isn’t smooth.”
“Then I’ll draw a squiggle between them.”
“That won’t work.” I’ll say.
“Why not?” You’ll ask.
“Because you’re really bad at drawing squiggles.” I’ll say.
If we want time to stay smooth, we need a special squiggle. Mathematicians use derivatives to make functions smooth. I don’t know why we’re asking mathematicians for advice on being smooth, but we’ll go with it. A smooth squiggle matches the lines at the first derivative and the second derivative and the third derivative and the fourth derivative and the fifth derivative…
Very smooth squiggles take a lot of math, but we’ll just skip to the end if that’s alright. I can’t be bothered to explain why it works. If you want to see the formula for time and travel, here it is. Otherwise, I drew you another picture.
$$P\left(x\right)=a\int_{s}^{x}e^{\frac{1}{\left(s-x\right)\left(d-x\right)}}dt+p_{1}\left(x\right)$$
So, there you have it. Time and travel. Bust out the graphing calculator next time you’re on a plane and blow everyone’s mind with this. Or maybe don’t. Either way, time will continue moving forward with a uniform velocity from the cradle to the grave.